Integrand size = 21, antiderivative size = 136 \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {3 b d^2 x}{4 c}-\frac {1}{3} i b d^2 x^2+\frac {1}{12} b c d^2 x^3+\frac {3 b d^2 \arctan (c x)}{4 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))+\frac {i b d^2 \log \left (1+c^2 x^2\right )}{3 c^2} \]
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Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {45, 4992, 12, 1816, 649, 209, 266} \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {3 b d^2 \arctan (c x)}{4 c^2}+\frac {i b d^2 \log \left (c^2 x^2+1\right )}{3 c^2}+\frac {1}{12} b c d^2 x^3-\frac {3 b d^2 x}{4 c}-\frac {1}{3} i b d^2 x^2 \]
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Rule 12
Rule 45
Rule 209
Rule 266
Rule 649
Rule 1816
Rule 4992
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))-(b c) \int \frac {d^2 x^2 \left (6+8 i c x-3 c^2 x^2\right )}{12 \left (1+c^2 x^2\right )} \, dx \\ & = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))-\frac {1}{12} \left (b c d^2\right ) \int \frac {x^2 \left (6+8 i c x-3 c^2 x^2\right )}{1+c^2 x^2} \, dx \\ & = \frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))-\frac {1}{12} \left (b c d^2\right ) \int \left (\frac {9}{c^2}+\frac {8 i x}{c}-3 x^2+\frac {i (9 i-8 c x)}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = -\frac {3 b d^2 x}{4 c}-\frac {1}{3} i b d^2 x^2+\frac {1}{12} b c d^2 x^3+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))-\frac {\left (i b d^2\right ) \int \frac {9 i-8 c x}{1+c^2 x^2} \, dx}{12 c} \\ & = -\frac {3 b d^2 x}{4 c}-\frac {1}{3} i b d^2 x^2+\frac {1}{12} b c d^2 x^3+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))+\frac {1}{3} \left (2 i b d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {\left (3 b d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 c} \\ & = -\frac {3 b d^2 x}{4 c}-\frac {1}{3} i b d^2 x^2+\frac {1}{12} b c d^2 x^3+\frac {3 b d^2 \arctan (c x)}{4 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))+\frac {2}{3} i c d^2 x^3 (a+b \arctan (c x))-\frac {1}{4} c^2 d^2 x^4 (a+b \arctan (c x))+\frac {i b d^2 \log \left (1+c^2 x^2\right )}{3 c^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.74 \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=\frac {d^2 \left (c x \left (a c x \left (6+8 i c x-3 c^2 x^2\right )+b \left (-9-4 i c x+c^2 x^2\right )\right )+b \left (9+6 c^2 x^2+8 i c^3 x^3-3 c^4 x^4\right ) \arctan (c x)+4 i b \log \left (1+c^2 x^2\right )\right )}{12 c^2} \]
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Time = 1.61 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84
method | result | size |
parts | \(a \,d^{2} \left (-\frac {1}{4} c^{2} x^{4}+\frac {2}{3} i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {b \,d^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(114\) |
derivativedivides | \(\frac {a \,d^{2} \left (-\frac {1}{4} c^{4} x^{4}+\frac {2}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+b \,d^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(120\) |
default | \(\frac {a \,d^{2} \left (-\frac {1}{4} c^{4} x^{4}+\frac {2}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+b \,d^{2} \left (-\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {2 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {3 c x}{4}+\frac {c^{3} x^{3}}{12}-\frac {i c^{2} x^{2}}{3}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(120\) |
parallelrisch | \(\frac {-3 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{2}+8 i x^{3} \arctan \left (c x \right ) b \,c^{3} d^{2}-3 a \,c^{4} d^{2} x^{4}+8 i x^{3} a \,c^{3} d^{2}+b \,c^{3} d^{2} x^{3}-4 i x^{2} b \,c^{2} d^{2}+6 x^{2} \arctan \left (c x \right ) b \,c^{2} d^{2}+6 x^{2} d^{2} c^{2} a +4 i b \,d^{2} \ln \left (c^{2} x^{2}+1\right )-9 b c \,d^{2} x +9 b \arctan \left (c x \right ) d^{2}}{12 c^{2}}\) | \(152\) |
risch | \(\frac {i d^{2} b \left (3 c^{2} x^{4}-8 i c \,x^{3}-6 x^{2}\right ) \ln \left (i c x +1\right )}{24}-\frac {x^{4} d^{2} c^{2} a}{4}-\frac {i d^{2} c^{2} x^{4} b \ln \left (-i c x +1\right )}{8}-\frac {d^{2} c b \,x^{3} \ln \left (-i c x +1\right )}{3}+\frac {x^{3} d^{2} c b}{12}+\frac {2 i a c \,d^{2} x^{3}}{3}+\frac {x^{2} d^{2} a}{2}+\frac {i d^{2} x^{2} b \ln \left (-i c x +1\right )}{4}-\frac {i b \,d^{2} x^{2}}{3}-\frac {3 b \,d^{2} x}{4 c}+\frac {3 b \,d^{2} \arctan \left (c x \right )}{4 c^{2}}+\frac {i d^{2} b \ln \left (81 c^{2} x^{2}+81\right )}{3 c^{2}}\) | \(191\) |
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Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.09 \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {6 \, a c^{4} d^{2} x^{4} + 2 \, {\left (-8 i \, a - b\right )} c^{3} d^{2} x^{3} - 4 \, {\left (3 \, a - 2 i \, b\right )} c^{2} d^{2} x^{2} + 18 \, b c d^{2} x - 17 i \, b d^{2} \log \left (\frac {c x + i}{c}\right ) + i \, b d^{2} \log \left (\frac {c x - i}{c}\right ) - {\left (-3 i \, b c^{4} d^{2} x^{4} - 8 \, b c^{3} d^{2} x^{3} + 6 i \, b c^{2} d^{2} x^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{24 \, c^{2}} \]
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Time = 1.84 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.76 \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=- \frac {a c^{2} d^{2} x^{4}}{4} - \frac {3 b d^{2} x}{4 c} - \frac {b d^{2} \left (\frac {i \log {\left (67 b c d^{2} x - 67 i b d^{2} \right )}}{24} - \frac {31 i \log {\left (67 b c d^{2} x + 67 i b d^{2} \right )}}{60}\right )}{c^{2}} - x^{3} \left (- \frac {2 i a c d^{2}}{3} - \frac {b c d^{2}}{12}\right ) - x^{2} \left (- \frac {a d^{2}}{2} + \frac {i b d^{2}}{3}\right ) + \left (\frac {i b c^{2} d^{2} x^{4}}{8} + \frac {b c d^{2} x^{3}}{3} - \frac {i b d^{2} x^{2}}{4}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (- 15 i b c^{4} d^{2} x^{4} - 40 b c^{3} d^{2} x^{3} + 30 i b c^{2} d^{2} x^{2} + 23 i b d^{2}\right ) \log {\left (- i c x + 1 \right )}}{120 c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.14 \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {1}{4} \, a c^{2} d^{2} x^{4} + \frac {2}{3} i \, a c d^{2} x^{3} - \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c^{2} d^{2} + \frac {1}{3} i \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b c d^{2} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} \]
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\[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )} x \,d x } \]
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Time = 0.75 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int x (d+i c d x)^2 (a+b \arctan (c x)) \, dx=\frac {\frac {d^2\,\left (9\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,4{}\mathrm {i}\right )}{12}-\frac {3\,b\,c\,d^2\,x}{4}}{c^2}+\frac {d^2\,\left (6\,a\,x^2+6\,b\,x^2\,\mathrm {atan}\left (c\,x\right )-b\,x^2\,4{}\mathrm {i}\right )}{12}-\frac {c^2\,d^2\,\left (3\,a\,x^4+3\,b\,x^4\,\mathrm {atan}\left (c\,x\right )\right )}{12}+\frac {c\,d^2\,\left (a\,x^3\,8{}\mathrm {i}+b\,x^3+b\,x^3\,\mathrm {atan}\left (c\,x\right )\,8{}\mathrm {i}\right )}{12} \]
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